On indecomposable elements of \(K_1\) of a product of elliptic curves
The author studies the kernel of the canonical map of Chow groups \(\text{CH}^2(X)\to \text{CH}^2 (E_1\times E_x)\), where \(E_1,E_2\) are elliptic curves with a good reduction over a local field of residue characteristic \(p\), \(X\) is a smooth projective model of \(E_1\times E_2\) over the ring of integers. The method is based on a construction of new indecomposable elements of \(H_{\text{Zar}}^1 (E_1\times E_2, {\mathcal K}_2)\). \({\mathcal K}_2\) denotes here an appropriate \(K\)-theory sheaf. The main theorem of the paper is the following: Theorem 3.1. Let \(k/\mathbb{Q}_p\) be a finite extension and let \(E_1, E_2\) be elliptic curves over \(k\) with good reduction. Let \(E_1,E_2\) be their smooth projective models over the ring of integers \(O\) of \(k\) and let \(X= E_1\times _{\text{Spec}(O)} E_2\). Then \(\Sigma= \text{ker} (\text{CH}^2(X)\to \text{CH}^2 (E_1\times E_2))\) is a finite \(p\)-group. The construction of the indecomposable elements relies on the work of \textit{G. Frey} and \textit{E. Kani} [in: Arithmetic algebraic geometry, Proc. Conf., Texel 1989, Prog. Math. 89, 153-176 (1991; Zbl 0757.14015)]. -- In section 2 the author reviews relevant constructions. Section 3 is devoted to the proof of the main result. Section 4 contains some applications of the theorem. It is worth noting that construction of new elements in \(H_{\text{Zar}}^1 (X,{\mathcal K}_2)\) is difficult and was previously considered by \textit{M. Flach} and independently by \textit{S. Mildenhall} in the case of a selfproduct of a modular elliptic curve.
scientific article; zbMATH DE number 1348189